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Crapo's Lemma states:

Let $X$ be a subset of a lattice $L$, and let $n_k$ be the number of $k$-element subsets of $X$ with join equal to $\hat{1}$ and meet equal to $\hat{0}$. Then $$\sum_k (-1)^{k}n_k=-\mu(\hat{0},\hat{1})+\sum_{x\le y,\ [x,y]\cap X=\varnothing}\mu(\hat{0},x)\mu(y,\hat{1}) .$$

I want to prove it with Crosscut Theorem: Let $L$ be a finite lattice and $X$ be a subset of $L$ s.t. (a) $\hat{1}\not\in X$ and (b) if $s\in L$ and $s\neq \hat{1}$, then $s\le t$ for some $t\in X$. Then $\sum_k(-1)^kN_k=\mu(\hat{0},\hat{1})$, where $N_k$ is the number of $k$-subsets of $X$ whose meet is $\hat{0}$.
It seems they have some similar thing on the left-hand set.

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